Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | #1039⋊(C43⋊C3) |
| derived, Hall | C52×C43 ⋊ C3×C4 |
| Hall | C43×#1039 ⋊ C3 |
| Hall | C52⋊(C43⋊C3) ⋊ C4 |
| Hall | #1039 ⋊ C43⋊C3 |
| Sylow | C43 ⋊ #1039⋊C3 |
| Sylow | C52 ⋊ C4×C43⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (C5⋊D5)⋊(C43⋊C3) |
| commutator, hypocenter | C52×C43 | C3×C4 |
| Fitting, socle, cosocle | C2×C52×C43 | C2×C3 |
| prime radical | C2×C52⋊(C43⋊C3) | C2 |
| LCS | G ▷ C52×C43 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 62 |
| transitive degree | μt(G) = 2580 |
A minimal faithful permutation representation — 2 generators on 62 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,33)(18,26,31)(21,29,28)(23,32,25)(27,36,40)(30,37,39)(34,41,35)(38,42,43)(44,45,47)(46,56,52,50,49,53)(48,58,57,54,55,51)(59,60,61,62);(1,2,4,8,15,20,5,9,16,25,35,32,12,19,28,24,34,41,22,31,39,36,14,23,33,40,43,42,26,7,13,21,30,38,37,17,3,6,11,18,27,29,10)(44,46,51,57,50)(45,48,49,56,54)(47,52,58,55,53)Polycyclic presentation:
2,3,5^2,43,2|a^2=f,c^a=c^4,d^a=d^4,c^b=cd^3,d^b=c^4d^3,e^b=e^36412 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 51);